The Central Angle Theorem can be extended to any pair of intersecting chords, not just those that happen to intersect on the circle.

If the chords intersect inside the circle, and are subtended by arcs of x and y, then the angle formed by the intersection of the chords is

(1/2) (x+y)

If the chords intersect outside the circle, and are subtended by arcs of x and y, with x > y, then the angle formed by the intersection of the chords is

(1/2) (x-y)

You can think of these formulas as equivalent if you imagine the signed arc length being negative if the arc bows in toward the vertex of the angle formed by the intersecting chords, and the signed arc length being positive if it bows outwards.   In that case, the angle formed by the intersection of the chords is always

(1/2) (x+y)

 . . . . . .  Some diagrams might be nice

Related pages in this website:

Summary of geometrical theorems

Central Angle Theorem (a.k.a. Inscribed angle property) is a special case of the Intersecting Chords Theorem.

The proofs of these theorems use the Inscribed Angle property of circles:

Law of Sines - Given triangle ABC with opposite sides a, b, and c, a/(sin A) = b/(sin B) = c/(sin C) = the diameter of the circumscribed circle.

Circumscribed Circle - The radius of a circle circumscribed around a triangle is R = abc/(4K), where K is the area of the triangle.

Cyclic Quadrilateral

Ptolemy's Theorem uses the facts presented here.

The Median and Altitude of a Right Triangle are reflections about the Right Angle Bisector

Internet References

Cut-the-knot: Angles In Circle 

The webmaster and author of the Math Help site is Graeme McRae.
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